Problem: Simplify the following expression: $q = \dfrac{9t^2 + 45t - 54}{t + 6} $
First factor the polynomial in the numerator. We notice that all the terms in the numerator have a common factor of $9$ , so we can rewrite the expression: $ q =\dfrac{9(t^2 + 5t - 6)}{t + 6} $ Then we factor the remaining polynomial: $t^2 + {5}t {-6} $ ${6} {-1} = {5}$ ${6} \times {-1} = {-6}$ $ (t + {6}) (t {-1}) $ This gives us a factored expression: $\dfrac{9(t + {6}) (t {-1})}{t + 6}$ We can divide the numerator and denominator by $(t - 6)$ on condition that $t \neq -6$ Therefore $q = 9(t - 1); t \neq -6$